### 2006-40

#### A note on implicit $\theta$-schemes applied on the Navier-Stokes-equations

by Rang, J..

Series: 2006-40, Preprints

MSC:
76D05 ~Navier-Stokes equations
35Q30 ~Navier-Stokes equations
65L20 Stability and convergence of numerical methods
65L60 Finite elements, ~Rayleigh-Ritz, Galerkin and collocation methods

Abstract:
In this note second order one-step- and fractional-step-$\theta$-schemes are applied on the
semidiscretised Navier-Stokes-equations. Both methods are formulated as Runge-Kutta-methods and are analysed. It is shown that the fractional-step-$\theta$-schemes have only stage order $q=1$ whereas the Crank-Nicolson-scheme has stage order $q=2$. Hence the fractional-step-$\theta$-scheme may have order reduction, if the method is applied on stiff ODEs and DAEs, i.e. the semi-discretised Navier-Stokes equations. Some theoretical results and numerical examples illustrate this phenomena. Moreover it is shown that there exists no fractional-step-$\theta$-method which has the stage order $q=2$ and is strongly A-stable.

Keywords:
imcompressible Navier-Stokes equations, implicit $\theta$-schemes, Runge-Kutta-methods, order reduction